The two-round system (also known as the second ballot, runoff voting or ballotage) is a voting system used to elect a single winner where the voter casts a single vote for their chosen candidate. However, if no candidate receives the required number of votes (usually an absolute majority or 40-45% with a winning margin of 5-15%), then those candidates having less than a certain proportion of the votes, or all but the two candidates receiving the most votes, are eliminated, and a second round of voting occurs.

The two-round system is known as run-off voting in the United States, where the second round is known as a run-off election. Runoff voting is also sometimes used as a generic term to describe any system involving a number of rounds of voting, with eliminations after each round. By this broader definition the two-round system is not the only form of run-off voting, and others include the exhaustive ballot and instant run-off voting (also known as the alternative vote). However the subject of this article is the two round system.

In Canada, for example, candidates for party leadership, when there are more than two, use an exhaustive ballot system, often called a run-off voting system. It is like a run-off system, except the one candidate must win a simple majority (over half). Candidates with the fewest votes or candidates who want to move their support to other candidates may also move to remove themselves from the next vote.

In both rounds of an election conducted using runoff voting, the voter simply marks an "X" beside his/her favorite candidate. If no candidate has an absolute majority of votes (i.e. more than half) in the first round, then the two candidates with the most votes proceed to a second round, from which all others are excluded. In the second round, because there are only two candidates, one candidate will achieve an absolute majority. In the second round each voter is entirely free to change the candidate he votes for, even if his preferred candidate has not yet been eliminated but he has merely changed his mind.

Some variants of the two round system use a different rule for choosing candidates for the second round, and allow more than two candidates to proceed to the second round. Under these systems it is sufficient for a candidate to receive a plurality of votes (i.e. more votes than anyone else) to be elected in the second round. In elections for the French National Assembly any candidate with fewer than 12.5% of the total vote is eliminated in the first round, and all remaining candidates are permitted to stand in the second round, in which a plurality is sufficient to be elected. Under some variants of runoff voting there is no formal rule for eliminating candidates, but, rather, candidates who receive few votes in the first round are expected to withdraw voluntarily. Historically, the President of Weimar Germany was popularly elected in 1925 and 1932 by a two-round system that in the second round allowed any candidate to run and did not require an absolute majority. In both elections the Communist candidate Ernst Thälmann did not withdraw and ran in the second round; in 1925 this probably ensured the election of Paul von Hindenburg (with only 48.3% of the vote), rather than Wilhelm Marx, the candidate of the centre parties.

This system can be coupled with a vote par circonscriptions when an area can provide multiple seats. For example, all French departements and territories, but the five less populated ones, have several deputies, like the 18 of Paris. French Polynesia, where most population is located on two islands, have 3 deputies at French parlement. The French Polynesia is so split in three sub-divided constituencies for this specific election. But this sub-divisions, aside from this election, have no reality and are not an administrative subdivision.

Imagine an election to choose which food to eat for dessert. There are 25 people having dessert and four candidates: Ice Cream, Apple Pie, Fruit and Celery. Runoff voting is used to find the winner.

Round 1: In the first round of voting each diner votes for the one candidate they most prefer. The results are as follows:

Ice Cream: 10 votes

Apple Pie: 6 votes

Fruit: 8 votes

Celery: 1 vote

Round 2: No candidate has an absolute majority of votes (in this election that would be 13) so the two candidates with the most votes, Ice Cream and Fruit, proceed to a second round, while Apple Pie and Celery are eliminated. Because their favorite candidates have been eliminated Apple Pie and Celery supporters must now vote for one of the two remaining candidates. The sole Celery supporter is health conscious, so now gives his vote to Fruit. However Apple Pie supporters are split: 3 prefer Ice Cream and 3 vote for Fruit. Of those who supported Ice Cream and Fruit in the first round no-one decides to change their vote. The results of the second round are therefore:

Ice Cream: 13

Fruit: 12

Result: Ice Cream now has an absolute majority so is declared the winner.

Imagine that the population of Tennessee, a state in the United States, is voting on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that they would all like the capital to be established as close to their own city as possible.

The candidates for the capital are:

Memphis, the state's largest city, with 42% of the voters, but located far from the other cities

Nashville, with 26% of the voters

Knoxville, with 17% of the voters

Chattanooga, with 15% of the voters

Round 1: In the first round of voting the results will be as follows:

Memphis: 42%

Nashville: 26%

Knoxville: 17%

Chattanooga: 15%

Round 2: No candidate has an absolute majority in the first round (this would be greater than 50%), so Memphis and Nashville proceed to the next round, while Knoxville and Chattanooga are excluded. Both eliminated cities are closer to Nashville than they are to Memphis. Therefore all of those who vote for either of the eliminated cities chose to vote for Nashville in the second round. None of the Memphis or Nashville supporters change their votes. The results are therefore:

Nashville: 58%

Memphis: 42%

Result: After round two Nashville has an absolute majority and is the winner. Note that while Nashville received only 26% of the votes and was significantly behind Memphis in the first round, it won the election by winning the second round.

This example demonstrates how the first two candidates from the first round might not necessarily be those expected to be the most popular by virtue of their party affiliations.

Several other candidates received smaller percentages of the first round vote.

Since no candidate obtained an absolute majority, a second round is organized with only the first two candidates from the first. Jean-Marie Le Pen, however, was widely recognized as a controversial nationalist candidate, and in the second round a vast majority of the votes went to Chirac:

The exhaustive ballot (EB) is similar to the two round system, but involves more rounds of voting rather than just two. If no candidate receives an absolute majority in the first round then the candidate(s) with the fewest number of votes is eliminated and excluded from further ballots. The process of exclusion and reballot continues until one candidate has an absolute majority. Because voters may have to cast votes several times, EB is not used in large-scale public elections. Instead it is used in smaller contests such as the election of the presiding officer of an assembly; one long-standing example of its use is in the United Kingdom, where local associations (LCAs) of the Conservative Party use EB to elect their prospective parliamentary candidates (PPCs). EB often elects a different winner from runoff voting. Because the two round system excludes more than one candidate after the first round, it is possible for a candidate to be eliminated who would have gone on to win the election under EB.

Instant-runoff voting (IRV) (also known as Preferential voting or Alternative Vote (AV)) like the exhaustive ballot involves multiple reiterative counts in which the candidate with fewest votes is eliminated each time. Whilst the exhaustive ballot and the two round system both involve voters casting a separate vote in each round, under instant-runoff voters vote only once. This is possible because, rather than voting for only a single candidate, the voter ranks all of the candidates in order of preference. These preferences are then used to "transfer" the votes of those whose first preference has been eliminated during the course of the count. Because the two round system and the exhaustive ballot involve separate rounds of voting, voters can use the results of one round to decide how they will vote in the next, whereas this is not possible under IRV. Because it is necessary only to vote once, IRV, like the two round system, is used for large-scale elections in many places. IRV often elects a different winner to the two round system and tends to produce the same results as the exhaustive ballot.

Variants of Instant Runoff voting can be designed to reflect the same rules as a two-round voting system. If no single candidate has an absolute majority of votes then only the two highest polling candidates progress to the second count all other candidates are excluded and their votes redistributed according to the recorded preferences for continuing candidates. One variant that works this way is called the Contingent vote, detailed below.

The Contingent vote is a variant of instant-runoff voting that has been used in the past in Queensland, in Australia. Under the contingent vote voters cast only one vote, by ranking all of the candidates in order of preference. However it involves only two rounds of counting and uses the same rule for eliminating candidates as the two round system. After the first round all but the two candidates with most votes are eliminated. Therefore one candidate always achieves an absolute majority in the second round. Because of these similarities the contingent vote tends to elect the same winner as the two round system, and often produces different results to instant-runoff voting. A variant of the contingent vote, called the supplementary vote, is used to elect some mayors in the United Kingdom. Another variant elects the President of Sri Lanka. A criticism of this method is that "it requires two polls, and gives opportunity for intrigue of various kinds."[1]

In the United States, the nonpartisan blanket primary, introduced in Louisiana for partisan state elections in 1975 and federal elections in 1978 (with a short return to a closed primary system in 2010), is virtually identical to the two-round system. Instead of the standard American system of primary elections to choose each party's candidate, followed by a general election contest between the winners of the primaries, the Louisiana primary allows voters to select any candidate, regardless of party affiliation. The two highest vote-winners in the primary—in effect, the first round of a two-round system—are then the only candidates whose names appear on the ballot at the general election, effectively requiring one candidate to win an absolute majority to take office.

The main difference between a nonpartisan blanket primary and a standard two-round system is that a second round of voting is required, even if one candidate wins an absolute majority of votes in the primary.

Most of the mathematical criteria by which voting systems are compared were formulated for voters with ordinal preferences. Some methods, like Approval voting, requests information than can't be unambiguously inferred from a single set of ordinal preferences. The two-round system is such a method, because the voters are not forced to vote according to a single ordinal preference in both rounds.

Since the two-round system requires more information from each voter than a single ordinal ballot provides, one can't fit the criteria that are formulated expressly for voters with ordinal preferences without making a generalization as to how the voters will behave. The same problem exists in Approval voting, where one has to make assumptions as to how the voters will place their approval cutoffs.

If the voters determine their preferences before the election and always vote directly consistent to them, they will emulate the Contingent vote and get the same results as if they were to use that system. Therefore, in that model of voting behavior, the two-round system passes all criteria that the contingent vote passes, and fails all criteria the contingent vote fails.

Since the voters in the two-round system don't have to choose their second round votes while voting in the first round, they are able to adjust their votes as players in a game. More complex models consider voter behavior when the voters reach a game-theoretical equilibrium from which they have no incentive, as defined by their internal preferences, to further change their behavior. However, because these equilibria are complex, only partial results are known. With respect to the voters' internal preferences, the two-round system passes the majority criterion in this model, as a majority can always coordinate to elect their preferred candidate. Also, in the case of three candidates or less and a robust political equilibrium,[2] the two-round system will pick the Condorcet winner whenever there is one, which is not the case in the Contingent vote model.

The equilibrium mentioned above is a perfect-information equilibrium and so only strictly holds in idealized conditions where every voter knows every other voter's preference. Thus it provides an upper bound on what can be achieved with rational (self-interested) coordination or knowledge of others' preferences. Since the voters almost surely won't have perfect information, it may not apply to real elections. In that matter, it is similar to the perfect competition model sometimes used in economics. To the extent that real elections approach this upper bound, large elections would do so less so than small ones, because it's less likely that a large electorate has information about all the other voters than that a small electorate has.

Runoff voting is intended to reduce the potential for eliminating "wasted" votes by tactical voting. Under the "first past the post" (plurality) system voters are encouraged to vote tactically by voting for only one of the two leading candidates, because a vote for any other candidate will not affect the result. Under runoff voting this tactic, known as "compromising", is sometimes unnecessary because, even if a voter's favourite candidate is eliminated in the first round, they will still have an opportunity to influence the result of the election by voting for a more popular candidate in the second round. However the tactic of compromising can still be used in runoff voting because it is sometimes necessary to compromise as a way of influencing which two candidates will survive to the second round. In order to do this it is necessary to vote for one of the three leading candidates in the first round, just as in an election held under the plurality system it is necessary to vote for one of the two leading candidates.

Runoff voting is also vulnerable to another tactic called "push over".[citation needed] This is a tactic by which voters vote tactically for an unpopular "push over" candidate in the first round as a way of helping their true favourite candidate win in the second round. The purpose of voting for the "push over", in theory, is to ensure that it is this weak candidate, rather than a stronger rival, who survives to challenge a one's preferred candidate in the second round. But in practice, such a tactic may prove counter-productive. If so many voters give their first preferences to the "weak" candidate that it ends up winning the first round, it is highly likely they will gain enough campaign momentum to have a strong chance of winning the runoff, too, and with it, the election. At the very least, their opponent would have to start taking the so-called "weak" candidate seriously, particularly if the runoff follows quickly after the first round.

Runoff voting can be influenced by strategic nomination; this is where candidates and political factions influence the result of an election by either nominating extra candidates or withdrawing a candidate who would otherwise have stood. Runoff voting is vulnerable to strategic nomination for the same reasons that it is open to the voting tactic of "compromising". This is because a candidate who knows they are unlikely to win can ensure that another candidate they support makes it to the second round by withdrawing from the race before the first round occurs, or by never choosing to stand in the first place. By withdrawing candidates a political faction can avoid the "spoiler effect", whereby a candidate "splits the vote" of its supporters. A famous example of this spoiler effect occurred in the 2002 French presidential election, when so many left-wing candidates stood in the first round that all of them were eliminated and two right-wing candidates advanced to the second round. Conversely, an important faction may have an interest in helping fund the campaign of smaller factions with a very different political agenda, so that these smaller parties end up weakening their own agenda.

Runoff voting encourages candidates to appeal to a broad cross-section of voters. This is because, in order to win an absolute majority in the second round, it is necessary for a candidate to win the support of voters whose favourite candidate has been eliminated. Under runoff voting, between rounds of voting, eliminated candidates, and the factions who previously supported them, often issue recommendations to their supporters as to whom to vote for in the second round of the contest. This means that eliminated candidates are still able to influence the result of the election. This influence leads to political bargaining between the two remaining candidates and the parties and candidates who have been eliminated, sometimes resulting in the two successful candidates making policy concessions to the less successful ones. Because it encourages conciliation and negotiation in these ways, runoff voting is advocated, in various forms, by some supporters of deliberative democracy.

Runoff voting is designed for single-seat constituencies. Therefore, like other single-seat methods, if used to elect a council or legislature it will not produce proportional representation (PR). This means that it is likely to lead to the representation of a small number of larger parties in an assembly, rather than a proliferation of small parties. In practice, runoff voting produces results very similar to those produced by the plurality system, and encourages a two-party system similar to those found in many countries that use plurality. Under a parliamentary system, it is more likely to produce single-party governments than are PR systems, which tend to produce coalition governments. While runoff voting is designed to ensure that each individual candidate elected is supported by a majority of those in their constituency, if used to elect an assembly it does not ensure this result on a national level. As in other non-PR systems, the party or coalition which wins a majority of seats will often not have the support of an absolute majority of voters across the nation.

The intention of runoff voting is that the winning candidate will have the support of an absolute majority of voters. Under the "first past the post" system the candidate with most votes (a plurality) wins, even if they do not have an absolute majority (more than half) of votes. The two rounds system tries to overcome this problem by permitting only two candidates in the second round, so that one must receive an absolute majority of votes.

Critics argue that the absolute majority obtained by the winner of runoff voting is an artificial one. As seen above, instant-runoff voting and the exhaustive ballot are two other voting systems that create an absolute majority for one candidate by eliminating weaker candidates over multiple rounds. However, as noted above in cases where there are 3 or more strong candidates, runoff voting will sometimes produce an absolute majority for a different winner than the candidate elected by the other two.

Advocates of Condorcet methods argue that a candidate can claim to have majority support only if they are the "Condorcet winner" – that is, the candidate who would beat every other candidate in a series of one-on-one elections. In runoff voting the winning candidate is only matched, one-on-one, with one of the other candidates. When a Condorcet winner exists, he does not necessarily win a runoff election due to insufficient support in the first round.

Runoff advocates counter that voters first preference is more important than lower preferences because that's where voters are putting the most effort of decision and that, unlike Condorcet methods, runoffs require a high showing among the full field of choices in addition to a strong showing in the final head-to-head competition. Condorcet methods can allow candidates to win who have minimal first-choice support and can win largely on the compromise appeal of being ranked second or third by more voters.

In large-scale public elections the two rounds of runoff voting are held on separate days, and so involve voters going to the polls twice. In smaller elections, such as those in assemblies or private organisations, it is sometimes possible to conduct both rounds in quick succession. However the fact that it involves two rounds means that, for large elections, runoff voting is more expensive than some other electoral systems. It may also lead to voter fatigue and a reduced turn-out in the second round. In every French presidential election since 1974, voter turnout actually increases in the second round. In runoff voting, the counting of votes in each round is simple and occurs in the same way as under the plurality system. Ranked voting systems, such as instant-runoff voting, involve a longer, more complicated count.

One of the strongest criticisms against the two-round voting system is the cost required to conduct two ballots. The two-round voting system also has the potential to cause political instability between the two rounds of voting, adding further to the economic impact of the two-round electoral system. Under an instant run-off ballot system there is only one round. It is alleged[by whom?] that the results of the election are known in days as opposed to months depending on the size of the electorate, but this is not always the case depending on the ability to count the rounds of the more complex instant runoff voting.

Supporters of IRV have claimed that IRV can save money in reduced costs, but this has not always been the case. In the 2009 Minneapolis IRV election, the single IRV round cost $365,000 more (adjusted for inflation) than the two rounds (2005 primary and general election) that IRV was supposed to replace[citation needed]. Combined with the fact that there is no single election system for the national presidential election and therefore no single method for IRV, there would be no way to guarantee that a national IRV election would save money – in fact it might cost even more.

The possibly increased costs of a two-round system have to be weighed against a debate of possibly better quality, between two clear options.